H-Termination proof of /home/matraf/haskell/eval_FullyBlown

YES 1.052 H-Termination proof of /home/matraf/haskell/eval_FullyBlown_Fast/empty.hs
H-Termination of the given Haskell-Program with start terms could successfully be proven:

↳ HASKELL

  ↳ BR

mainModule Main

((minimum :: [Char] -> Char) :: [Char] -> Char)

module Main where

import qualified Prelude

Replaced joker patterns by fresh variables and removed binding patterns.

↳ HASKELL

  ↳ BR

    ↳ HASKELL

      ↳ COR

mainModule Main

((minimum :: [Char] -> Char) :: [Char] -> Char)

module Main where

import qualified Prelude

Cond Reductions:
The following Function with conditions

min x y

| x <= y

= x

| otherwise

= y

is transformed to

min x y = min2 x y

min0 x y True = y

min1 x y True = x

min1 x y False = min0 x y otherwise

min2 x y = min1 x y (x <= y)

The following Function with conditions

undefined

| False

= undefined

is transformed to

undefined = undefined1

undefined0 True = undefined

undefined1 = undefined0 False

↳ HASKELL

  ↳ BR

    ↳ HASKELL

      ↳ COR

        ↳ HASKELL

          ↳ Narrow

mainModule Main

(minimum :: [Char] -> Char)

module Main where

import qualified Prelude

Haskell To QDPs

digraph dp_graph {
node [outthreshold=100, inthreshold=100];1[label="minimum",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3];
3[label="minimum vx3",fontsize=16,color="black",shape="triangle"];3 -> 4[label="",style="solid", color="black", weight=3];
4[label="foldl1 min vx3",fontsize=16,color="burlywood",shape="box"];312[label="vx3/vx30 : vx31",fontsize=10,color="white",style="solid",shape="box"];4 -> 312[label="",style="solid", color="burlywood", weight=9];
312 -> 5[label="",style="solid", color="burlywood", weight=3];
313[label="vx3/[]",fontsize=10,color="white",style="solid",shape="box"];4 -> 313[label="",style="solid", color="burlywood", weight=9];
313 -> 6[label="",style="solid", color="burlywood", weight=3];
5[label="foldl1 min (vx30 : vx31)",fontsize=16,color="black",shape="box"];5 -> 7[label="",style="solid", color="black", weight=3];
6[label="foldl1 min []",fontsize=16,color="black",shape="box"];6 -> 8[label="",style="solid", color="black", weight=3];
7[label="foldl min vx30 vx31",fontsize=16,color="burlywood",shape="triangle"];314[label="vx31/vx310 : vx311",fontsize=10,color="white",style="solid",shape="box"];7 -> 314[label="",style="solid", color="burlywood", weight=9];
314 -> 9[label="",style="solid", color="burlywood", weight=3];
315[label="vx31/[]",fontsize=10,color="white",style="solid",shape="box"];7 -> 315[label="",style="solid", color="burlywood", weight=9];
315 -> 10[label="",style="solid", color="burlywood", weight=3];
8[label="error []",fontsize=16,color="red",shape="box"];9[label="foldl min vx30 (vx310 : vx311)",fontsize=16,color="black",shape="box"];9 -> 11[label="",style="solid", color="black", weight=3];
10[label="foldl min vx30 []",fontsize=16,color="black",shape="box"];10 -> 12[label="",style="solid", color="black", weight=3];
11 -> 7[label="",style="dashed", color="red", weight=0];
11[label="foldl min (min vx30 vx310) vx311",fontsize=16,color="magenta"];11 -> 13[label="",style="dashed", color="magenta", weight=3];
11 -> 14[label="",style="dashed", color="magenta", weight=3];
12[label="vx30",fontsize=16,color="green",shape="box"];13[label="vx311",fontsize=16,color="green",shape="box"];14[label="min vx30 vx310",fontsize=16,color="black",shape="box"];14 -> 15[label="",style="solid", color="black", weight=3];
15[label="min2 vx30 vx310",fontsize=16,color="black",shape="box"];15 -> 16[label="",style="solid", color="black", weight=3];
16[label="min1 vx30 vx310 (vx30 <= vx310)",fontsize=16,color="black",shape="box"];16 -> 17[label="",style="solid", color="black", weight=3];
17[label="min1 vx30 vx310 (compare vx30 vx310 /= GT)",fontsize=16,color="black",shape="box"];17 -> 18[label="",style="solid", color="black", weight=3];
18[label="min1 vx30 vx310 (not (compare vx30 vx310 == GT))",fontsize=16,color="black",shape="box"];18 -> 19[label="",style="solid", color="black", weight=3];
19[label="min1 vx30 vx310 (not (primCmpChar vx30 vx310 == GT))",fontsize=16,color="burlywood",shape="box"];317[label="vx30/Char vx300",fontsize=10,color="white",style="solid",shape="box"];19 -> 317[label="",style="solid", color="burlywood", weight=9];
317 -> 20[label="",style="solid", color="burlywood", weight=3];
20[label="min1 (Char vx300) vx310 (not (primCmpChar (Char vx300) vx310 == GT))",fontsize=16,color="burlywood",shape="box"];318[label="vx310/Char vx3100",fontsize=10,color="white",style="solid",shape="box"];20 -> 318[label="",style="solid", color="burlywood", weight=9];
318 -> 21[label="",style="solid", color="burlywood", weight=3];
21[label="min1 (Char vx300) (Char vx3100) (not (primCmpChar (Char vx300) (Char vx3100) == GT))",fontsize=16,color="black",shape="box"];21 -> 22[label="",style="solid", color="black", weight=3];
22[label="min1 (Char vx300) (Char vx3100) (not (primCmpNat vx300 vx3100 == GT))",fontsize=16,color="burlywood",shape="box"];319[label="vx300/Succ vx3000",fontsize=10,color="white",style="solid",shape="box"];22 -> 319[label="",style="solid", color="burlywood", weight=9];
319 -> 23[label="",style="solid", color="burlywood", weight=3];
320[label="vx300/Zero",fontsize=10,color="white",style="solid",shape="box"];22 -> 320[label="",style="solid", color="burlywood", weight=9];
320 -> 24[label="",style="solid", color="burlywood", weight=3];
23[label="min1 (Char (Succ vx3000)) (Char vx3100) (not (primCmpNat (Succ vx3000) vx3100 == GT))",fontsize=16,color="burlywood",shape="box"];321[label="vx3100/Succ vx31000",fontsize=10,color="white",style="solid",shape="box"];23 -> 321[label="",style="solid", color="burlywood", weight=9];
321 -> 25[label="",style="solid", color="burlywood", weight=3];
322[label="vx3100/Zero",fontsize=10,color="white",style="solid",shape="box"];23 -> 322[label="",style="solid", color="burlywood", weight=9];
322 -> 26[label="",style="solid", color="burlywood", weight=3];
24[label="min1 (Char Zero) (Char vx3100) (not (primCmpNat Zero vx3100 == GT))",fontsize=16,color="burlywood",shape="box"];323[label="vx3100/Succ vx31000",fontsize=10,color="white",style="solid",shape="box"];24 -> 323[label="",style="solid", color="burlywood", weight=9];
323 -> 27[label="",style="solid", color="burlywood", weight=3];
324[label="vx3100/Zero",fontsize=10,color="white",style="solid",shape="box"];24 -> 324[label="",style="solid", color="burlywood", weight=9];
324 -> 28[label="",style="solid", color="burlywood", weight=3];
25[label="min1 (Char (Succ vx3000)) (Char (Succ vx31000)) (not (primCmpNat (Succ vx3000) (Succ vx31000) == GT))",fontsize=16,color="black",shape="box"];25 -> 29[label="",style="solid", color="black", weight=3];
26[label="min1 (Char (Succ vx3000)) (Char Zero) (not (primCmpNat (Succ vx3000) Zero == GT))",fontsize=16,color="black",shape="box"];26 -> 30[label="",style="solid", color="black", weight=3];
27[label="min1 (Char Zero) (Char (Succ vx31000)) (not (primCmpNat Zero (Succ vx31000) == GT))",fontsize=16,color="black",shape="box"];27 -> 31[label="",style="solid", color="black", weight=3];
28[label="min1 (Char Zero) (Char Zero) (not (primCmpNat Zero Zero == GT))",fontsize=16,color="black",shape="box"];28 -> 32[label="",style="solid", color="black", weight=3];
29 -> 250[label="",style="dashed", color="red", weight=0];
29[label="min1 (Char (Succ vx3000)) (Char (Succ vx31000)) (not (primCmpNat vx3000 vx31000 == GT))",fontsize=16,color="magenta"];29 -> 251[label="",style="dashed", color="magenta", weight=3];
29 -> 252[label="",style="dashed", color="magenta", weight=3];
29 -> 253[label="",style="dashed", color="magenta", weight=3];
29 -> 254[label="",style="dashed", color="magenta", weight=3];
30[label="min1 (Char (Succ vx3000)) (Char Zero) (not (GT == GT))",fontsize=16,color="black",shape="box"];30 -> 35[label="",style="solid", color="black", weight=3];
31[label="min1 (Char Zero) (Char (Succ vx31000)) (not (LT == GT))",fontsize=16,color="black",shape="box"];31 -> 36[label="",style="solid", color="black", weight=3];
32[label="min1 (Char Zero) (Char Zero) (not (EQ == GT))",fontsize=16,color="black",shape="box"];32 -> 37[label="",style="solid", color="black", weight=3];
251[label="vx31000",fontsize=16,color="green",shape="box"];252[label="vx3000",fontsize=16,color="green",shape="box"];253[label="vx3000",fontsize=16,color="green",shape="box"];254[label="vx31000",fontsize=16,color="green",shape="box"];250[label="min1 (Char (Succ vx26)) (Char (Succ vx27)) (not (primCmpNat vx28 vx29 == GT))",fontsize=16,color="burlywood",shape="triangle"];326[label="vx28/Succ vx280",fontsize=10,color="white",style="solid",shape="box"];250 -> 326[label="",style="solid", color="burlywood", weight=9];
326 -> 291[label="",style="solid", color="burlywood", weight=3];
327[label="vx28/Zero",fontsize=10,color="white",style="solid",shape="box"];250 -> 327[label="",style="solid", color="burlywood", weight=9];
327 -> 292[label="",style="solid", color="burlywood", weight=3];
35[label="min1 (Char (Succ vx3000)) (Char Zero) (not True)",fontsize=16,color="black",shape="box"];35 -> 42[label="",style="solid", color="black", weight=3];
36[label="min1 (Char Zero) (Char (Succ vx31000)) (not False)",fontsize=16,color="black",shape="box"];36 -> 43[label="",style="solid", color="black", weight=3];
37[label="min1 (Char Zero) (Char Zero) (not False)",fontsize=16,color="black",shape="box"];37 -> 44[label="",style="solid", color="black", weight=3];
291[label="min1 (Char (Succ vx26)) (Char (Succ vx27)) (not (primCmpNat (Succ vx280) vx29 == GT))",fontsize=16,color="burlywood",shape="box"];328[label="vx29/Succ vx290",fontsize=10,color="white",style="solid",shape="box"];291 -> 328[label="",style="solid", color="burlywood", weight=9];
328 -> 293[label="",style="solid", color="burlywood", weight=3];
329[label="vx29/Zero",fontsize=10,color="white",style="solid",shape="box"];291 -> 329[label="",style="solid", color="burlywood", weight=9];
329 -> 294[label="",style="solid", color="burlywood", weight=3];
292[label="min1 (Char (Succ vx26)) (Char (Succ vx27)) (not (primCmpNat Zero vx29 == GT))",fontsize=16,color="burlywood",shape="box"];330[label="vx29/Succ vx290",fontsize=10,color="white",style="solid",shape="box"];292 -> 330[label="",style="solid", color="burlywood", weight=9];
330 -> 295[label="",style="solid", color="burlywood", weight=3];
331[label="vx29/Zero",fontsize=10,color="white",style="solid",shape="box"];292 -> 331[label="",style="solid", color="burlywood", weight=9];
331 -> 296[label="",style="solid", color="burlywood", weight=3];
42[label="min1 (Char (Succ vx3000)) (Char Zero) False",fontsize=16,color="black",shape="box"];42 -> 49[label="",style="solid", color="black", weight=3];
43[label="min1 (Char Zero) (Char (Succ vx31000)) True",fontsize=16,color="black",shape="box"];43 -> 50[label="",style="solid", color="black", weight=3];
44[label="min1 (Char Zero) (Char Zero) True",fontsize=16,color="black",shape="box"];44 -> 51[label="",style="solid", color="black", weight=3];
293[label="min1 (Char (Succ vx26)) (Char (Succ vx27)) (not (primCmpNat (Succ vx280) (Succ vx290) == GT))",fontsize=16,color="black",shape="box"];293 -> 297[label="",style="solid", color="black", weight=3];
294[label="min1 (Char (Succ vx26)) (Char (Succ vx27)) (not (primCmpNat (Succ vx280) Zero == GT))",fontsize=16,color="black",shape="box"];294 -> 298[label="",style="solid", color="black", weight=3];
295[label="min1 (Char (Succ vx26)) (Char (Succ vx27)) (not (primCmpNat Zero (Succ vx290) == GT))",fontsize=16,color="black",shape="box"];295 -> 299[label="",style="solid", color="black", weight=3];
296[label="min1 (Char (Succ vx26)) (Char (Succ vx27)) (not (primCmpNat Zero Zero == GT))",fontsize=16,color="black",shape="box"];296 -> 300[label="",style="solid", color="black", weight=3];
49[label="min0 (Char (Succ vx3000)) (Char Zero) otherwise",fontsize=16,color="black",shape="box"];49 -> 57[label="",style="solid", color="black", weight=3];
50[label="Char Zero",fontsize=16,color="green",shape="box"];51[label="Char Zero",fontsize=16,color="green",shape="box"];297 -> 250[label="",style="dashed", color="red", weight=0];
297[label="min1 (Char (Succ vx26)) (Char (Succ vx27)) (not (primCmpNat vx280 vx290 == GT))",fontsize=16,color="magenta"];297 -> 301[label="",style="dashed", color="magenta", weight=3];
297 -> 302[label="",style="dashed", color="magenta", weight=3];
298[label="min1 (Char (Succ vx26)) (Char (Succ vx27)) (not (GT == GT))",fontsize=16,color="black",shape="box"];298 -> 303[label="",style="solid", color="black", weight=3];
299[label="min1 (Char (Succ vx26)) (Char (Succ vx27)) (not (LT == GT))",fontsize=16,color="black",shape="box"];299 -> 304[label="",style="solid", color="black", weight=3];
300[label="min1 (Char (Succ vx26)) (Char (Succ vx27)) (not (EQ == GT))",fontsize=16,color="black",shape="box"];300 -> 305[label="",style="solid", color="black", weight=3];
57[label="min0 (Char (Succ vx3000)) (Char Zero) True",fontsize=16,color="black",shape="box"];57 -> 65[label="",style="solid", color="black", weight=3];
301[label="vx290",fontsize=16,color="green",shape="box"];302[label="vx280",fontsize=16,color="green",shape="box"];303[label="min1 (Char (Succ vx26)) (Char (Succ vx27)) (not True)",fontsize=16,color="black",shape="box"];303 -> 306[label="",style="solid", color="black", weight=3];
304[label="min1 (Char (Succ vx26)) (Char (Succ vx27)) (not False)",fontsize=16,color="black",shape="triangle"];304 -> 307[label="",style="solid", color="black", weight=3];
305 -> 304[label="",style="dashed", color="red", weight=0];
305[label="min1 (Char (Succ vx26)) (Char (Succ vx27)) (not False)",fontsize=16,color="magenta"];65[label="Char Zero",fontsize=16,color="green",shape="box"];306[label="min1 (Char (Succ vx26)) (Char (Succ vx27)) False",fontsize=16,color="black",shape="box"];306 -> 308[label="",style="solid", color="black", weight=3];
307[label="min1 (Char (Succ vx26)) (Char (Succ vx27)) True",fontsize=16,color="black",shape="box"];307 -> 309[label="",style="solid", color="black", weight=3];
308[label="min0 (Char (Succ vx26)) (Char (Succ vx27)) otherwise",fontsize=16,color="black",shape="box"];308 -> 310[label="",style="solid", color="black", weight=3];
309[label="Char (Succ vx26)",fontsize=16,color="green",shape="box"];310[label="min0 (Char (Succ vx26)) (Char (Succ vx27)) True",fontsize=16,color="black",shape="box"];310 -> 311[label="",style="solid", color="black", weight=3];
311[label="Char (Succ vx27)",fontsize=16,color="green",shape="box"];}

↳ HASKELL

  ↳ BR

    ↳ HASKELL

      ↳ COR

        ↳ HASKELL

          ↳ Narrow

            ↳ AND

              ↳ QDP

                ↳ QDPSizeChangeProof

              ↳ QDP

Q DP problem:
The TRS P consists of the following rules:

new_min1(vx26, vx27, Succ(vx280), Succ(vx290)) → new_min1(vx26, vx27, vx280, vx290)

R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By using the subterm criterion [20] together with the size-change analysis [32] we have proven that there are no infinite chains for this DP problem.

From the DPs we obtained the following set of size-change graphs:

new_min1(vx26, vx27, Succ(vx280), Succ(vx290)) → new_min1(vx26, vx27, vx280, vx290)
The graph contains the following edges 1 >= 1, 2 >= 2, 3 > 3, 4 > 4

↳ HASKELL

  ↳ BR

    ↳ HASKELL

      ↳ COR

        ↳ HASKELL

          ↳ Narrow

            ↳ AND

              ↳ QDP

              ↳ QDP

                ↳ QDPSizeChangeProof

Q DP problem:
The TRS P consists of the following rules:

new_foldl(vx30, :(vx310, vx311)) → new_foldl(new_min10(vx30, vx310), vx311)

The TRS R consists of the following rules:

new_min11(vx26, vx27, Zero, Zero) → new_min12(vx26, vx27)
new_min10(Char(Succ(vx3000)), Char(Zero)) → Char(Zero)
new_min10(Char(Zero), Char(Succ(vx31000))) → Char(Zero)
new_min12(vx26, vx27) → Char(Succ(vx26))
new_min10(Char(Succ(vx3000)), Char(Succ(vx31000))) → new_min11(vx3000, vx31000, vx3000, vx31000)
new_min11(vx26, vx27, Succ(vx280), Zero) → Char(Succ(vx27))
new_min11(vx26, vx27, Succ(vx280), Succ(vx290)) → new_min11(vx26, vx27, vx280, vx290)
new_min11(vx26, vx27, Zero, Succ(vx290)) → new_min12(vx26, vx27)
new_min10(Char(Zero), Char(Zero)) → Char(Zero)

The set Q consists of the following terms:

new_min11(x0, x1, Zero, Succ(x2))
new_min10(Char(Zero), Char(Zero))
new_min10(Char(Succ(x0)), Char(Zero))
new_min11(x0, x1, Succ(x2), Zero)
new_min11(x0, x1, Succ(x2), Succ(x3))
new_min10(Char(Succ(x0)), Char(Succ(x1)))
new_min11(x0, x1, Zero, Zero)
new_min12(x0, x1)
new_min10(Char(Zero), Char(Succ(x0)))

We have to consider all minimal (P,Q,R)-chains.
By using the subterm criterion [20] together with the size-change analysis [32] we have proven that there are no infinite chains for this DP problem.

From the DPs we obtained the following set of size-change graphs:

new_foldl(vx30, :(vx310, vx311)) → new_foldl(new_min10(vx30, vx310), vx311)
The graph contains the following edges 2 > 2